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<span class="kw">let </span><font color="Maroon" title="c1">i</font>, <font color="Maroon" title="c2">m</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">f1</font>, <font color="Olive" title="b2">f2</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  <a href="relat_1.html#V4" title="RELAT_1:attr.4">-defined</a>   <a href="comput_1.html#NV2" title="COMPUT_1:NV.2">to-naturals</a>   <a href="unialg_1.html#V1" title="UNIALG_1:attr.1">homogeneous</a>  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><br/> for <font color="Olive" title="b3">p</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Olive" title="b1">f1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b3">p</font> holds <br/>( ( <font color="Olive" title="b3">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Olive" title="b1">f1</font>,<font color="Olive" title="b2">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> implies  <a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <font color="Olive" title="b3">p</font>,<font color="Maroon" title="c1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b1">f1</font> ) &amp; (  <a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <font color="Olive" title="b3">p</font>,<font color="Maroon" title="c1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b1">f1</font> implies <font color="Olive" title="b3">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Olive" title="b1">f1</font>,<font color="Olive" title="b2">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> ) &amp; ( <font color="Olive" title="b3">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Olive" title="b1">f1</font>,<font color="Olive" title="b2">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> implies <span class="p1">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Olive" title="b1">f1</font>,<font color="Olive" title="b2">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <font color="Olive" title="b3">p</font>,<font color="Maroon" title="c1">i</font></span>)</span> ) &amp; ( <font color="Olive" title="b3">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Olive" title="b1">f1</font>,<font color="Olive" title="b2">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> implies ( <font color="Olive" title="b3">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Olive" title="b1">f1</font>,<font color="Olive" title="b2">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Olive" title="b3">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Olive" title="b1">f1</font>,<font color="Olive" title="b2">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b3">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b2">f2</font> ) ) &amp; ( <font color="Olive" title="b3">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Olive" title="b1">f1</font>,<font color="Olive" title="b2">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Olive" title="b3">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Olive" title="b1">f1</font>,<font color="Olive" title="b2">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b3">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b2">f2</font> implies <font color="Olive" title="b3">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Olive" title="b1">f1</font>,<font color="Olive" title="b2">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> ) &amp; ( <font color="Olive" title="b3">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Olive" title="b1">f1</font>,<font color="Olive" title="b2">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> implies <span class="p1">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Olive" title="b1">f1</font>,<font color="Olive" title="b2">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b3">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Olive" title="b1">f1</font>,<font color="Olive" title="b2">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Olive" title="b3">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">f1</font>, <font color="Maroon" title="c4">f2</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  <a href="relat_1.html#V4" title="RELAT_1:attr.4">-defined</a>   <a href="comput_1.html#NV2" title="COMPUT_1:NV.2">to-naturals</a>   <a href="unialg_1.html#V1" title="UNIALG_1:attr.1">homogeneous</a>  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">p</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c3">f1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b1">p</font> holds <br/>( ( <font color="Olive" title="b1">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> implies  <a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <font color="Olive" title="b1">p</font>,<font color="Maroon" title="c1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">f1</font> ) &amp; (  <a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <font color="Olive" title="b1">p</font>,<font color="Maroon" title="c1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">f1</font> implies <font color="Olive" title="b1">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> ) &amp; ( <font color="Olive" title="b1">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> implies <span class="p1">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <font color="Olive" title="b1">p</font>,<font color="Maroon" title="c1">i</font></span>)</span> ) &amp; ( <font color="Olive" title="b1">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> implies ( <font color="Olive" title="b1">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">f2</font> ) ) &amp; ( <font color="Olive" title="b1">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">f2</font> implies <font color="Olive" title="b1">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> ) &amp; ( <font color="Olive" title="b1">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> implies <span class="p1">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Olive" title="b1">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">p</font> be    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c3">f1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">p</font> implies ( ( <font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> implies  <a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">f1</font> ) &amp; (  <a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">f1</font> implies <font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> ) &amp; ( <font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> implies <span class="p1">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c1">i</font></span>)</span> ) &amp; ( <font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> implies ( <font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">f2</font> ) ) &amp; ( <font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">f2</font> implies <font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> ) &amp; ( <font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> implies <span class="p1">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> ) ) )</span><br/>









<span class="kw">set </span><font color="Maroon" title="c6">p0</font> = <font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">G</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c3">f1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span>,<span class="p1">(<span class="default"><a href="comput_1.html#K2" title="COMPUT_1:func.2">HFuncs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span><span class="kw"> such that </span><br/><a NAME="E2:108"/><span class="lab"><font color="Green" title="E66">A1</font></span>: 
 <a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c7">G</font>
 <span class="kw">and </span><br/><a NAME="E3:108"/><span class="lab"><font color="Green" title="E67">A2</font></span>: 
for <font color="Olive" title="b1">p</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c3">f1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c7">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="comput_1.html#K6" title="COMPUT_1:func.6">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font>,<font color="Olive" title="b1">p</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="comput_1.html#D14" target="_self" title="COMPUT_1:def.14">Def14</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">rngG</font> =  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">G</font> as   <a href="funct_1.html#V4" title="FUNCT_1:attr.4">functional</a>   <a href="comput_1.html#V1" title="COMPUT_1:attr.1">compatible</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:108"><span class="lab"><font color="Green" title="E66">A1</font></span></a>, <a class="ref" href="comput_1.html#T14" target="_self" title="COMPUT_1:th.14">Th14</a></span>;<br/>
<span class="kw">assume </span><a NAME="E5:108"/><span class="lab"><font color="Green" title="E68">A4</font></span>: 
<font color="Maroon" title="c1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> implies  <a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">f1</font> ) &amp; (  <a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">f1</font> implies <font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> ) &amp; ( <font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> implies <span class="p1">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c1">i</font></span>)</span> ) &amp; ( <font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> implies ( <font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">f2</font> ) ) &amp; ( <font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">f2</font> implies <font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> ) &amp; ( <font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> implies <span class="p1">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> ) )</span><br/>

<div><span class="lab"><font color="Green" title="E69">A5</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="108_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c9">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c9">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> implies <font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c9">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c9">k</font></span>)</span></span>)</span> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:108_1"/><span class="lab"><font color="Green" title="E70">A6</font></span>: 
<font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c9">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E2:108_1"/><span class="lab"><font color="Green" title="E71">A7</font></span>: 
not <font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c9">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c9">k</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E3:108_1"/>
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c8">rngG</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:108"><span class="lab"><font color="Green" title="E66">A1</font></span></a>, <a class="txt" href="#E1:108_1"><span class="lab"><font color="Green" title="E70">A6</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">rngG</font> =  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">G</font> as  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="funct_1.html#V4" title="FUNCT_1:attr.4">functional</a>   <a href="comput_1.html#V1" title="COMPUT_1:attr.1">compatible</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/><span class="kw">set </span><font color="Maroon" title="c11">pk</font> = <font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c9">k</font>;<br/><a NAME="E5:108_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c10">rngG</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b1">f</font></span>)</span> where <font color="Olive" title="b1">f</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c10">rngG</font> : verum </span>}</span> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="comput_1.html#T15" target="_self" title="COMPUT_1:th.15">Th15</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">X</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E7:108_1"/><span class="lab"><font color="Green" title="E72">A8</font></span>: 
<font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c9">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">X</font>
 <span class="kw">and </span><br/><a NAME="E8:108_1"/><span class="lab"><font color="Green" title="E73">A9</font></span>: 
<font color="Maroon" title="c12">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b1">f</font></span>)</span> where <font color="Olive" title="b1">f</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c10">rngG</font> : verum </span>}</span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:108"><span class="lab"><font color="Green" title="E66">A1</font></span></a>, <a class="txt" href="#E1:108_1"><span class="lab"><font color="Green" title="E70">A6</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c13">f</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c10">rngG</font><span class="kw"> such that </span><br/><a NAME="E10:108_1"/><span class="lab"><font color="Green" title="E74">A10</font></span>: 
<font color="Maroon" title="c12">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c13">f</font>
 <span class="kw">and </span><br/><a NAME="E11:108_1"/>
verum
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:108_1"><span class="lab"><font color="Green" title="E73">A9</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c14">pp</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E13:108_1"/><span class="lab"><font color="Green" title="E75">A11</font></span>: 
<font color="Maroon" title="c14">pp</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">G</font>
 <span class="kw">and </span><br/><a NAME="E14:108_1"/><span class="lab"><font color="Green" title="E76">A12</font></span>: 
<font color="Maroon" title="c13">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">pp</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c15">pp</font> = <font color="Maroon" title="c14">pp</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c3">f1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:108_1"><span class="lab"><font color="Green" title="E75">A11</font></span></a></span>;<br/><a NAME="E16:108_1"/>
<font color="Maroon" title="c7">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">pp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="comput_1.html#K6" title="COMPUT_1:func.6">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c15">pp</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:108"><span class="lab"><font color="Green" title="E67">A2</font></span></a></span>;<br/><a NAME="E17:108_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A13</font></span>: 
 ex <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">pp</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Olive" title="b1">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:108_1"><span class="lab"><font color="Green" title="E72">A8</font></span></a>, <a class="txt" href="#E10:108_1"><span class="lab"><font color="Green" title="E74">A10</font></span></a>, <a class="txt" href="#E14:108_1"><span class="lab"><font color="Green" title="E76">A12</font></span></a>, <a class="ref" href="comput_1.html#T53" target="_self" title="COMPUT_1:th.53">Th53</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c16">ppi</font> = <font color="Maroon" title="c15">pp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c1">i</font>;<br/><span class="lab"><font color="Green" title="E78">A14</font></span>: <font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c15">pp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c1">i</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c9">k</font></span>)</span> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c15">pp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c1">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T36" title="FUNCT_7:th.36">FUNCT_7:36</a></span>
<br/>.= 
<font color="Maroon" title="c15">pp</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c15">pp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c1">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:108_1"><span class="lab"><font color="Green" title="E77">A13</font></span></a>, <a class="ref" href="funct_7.html#T36" title="FUNCT_7:th.36">FUNCT_7:36</a></span>
<br/>.= 
<font color="Maroon" title="c15">pp</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T37" title="FUNCT_7:th.37">FUNCT_7:37</a></span>
;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_1_2"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:108_1_2"/>
( <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">pp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c1">i</font> or <font color="Maroon" title="c15">pp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">k</font> or <font color="Maroon" title="c15">pp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c1">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c9">k</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_1_2_1"><span class="kw">suppose </span></a><a NAME="E1:108_1_2_1"/>
<font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">pp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c1">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:108_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:108_1"><span class="lab"><font color="Green" title="E71">A7</font></span></a>, <a class="txt" href="#E7:108_1"><span class="lab"><font color="Green" title="E72">A8</font></span></a>, <a class="txt" href="#E10:108_1"><span class="lab"><font color="Green" title="E74">A10</font></span></a>, <a class="txt" href="#E14:108_1"><span class="lab"><font color="Green" title="E76">A12</font></span></a>, <a class="txt" href="#E18:108_1"><span class="lab"><font color="Green" title="E78">A14</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_1_2_2"><span class="kw">suppose </span></a><a NAME="E1:108_1_2_2"/>
<font color="Maroon" title="c15">pp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c17">m</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E3:108_1_2_2"/><span class="lab"><font color="Green" title="E79">A15</font></span>: 
<font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">pp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c1">i</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c17">m</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T10" title="NAT_1:th.10">NAT_1:10</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c18">m</font> = <font color="Maroon" title="c17">m</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/><a NAME="E5:108_1_2_2"/>
<font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">pp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c1">i</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c18">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:108_1_2_2"><span class="lab"><font color="Green" title="E79">A15</font></span></a></span>;<br/><a NAME="E6:108_1_2_2"/><span class="kw">then </span>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">pp</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c9">k</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:108"><span class="lab"><font color="Green" title="E68">A4</font></span></a>, <a class="txt" href="#E3:108"><span class="lab"><font color="Green" title="E67">A2</font></span></a>, <a class="txt" href="#E18:108_1"><span class="lab"><font color="Green" title="E78">A14</font></span></a>, <a class="txt" href="comput_1.html#E79"><span class="lab"><font color="Green" title="E63">Lm4</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E7:108_1_2_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:108_1"><span class="lab"><font color="Green" title="E71">A7</font></span></a>, <a class="txt" href="#E7:108_1"><span class="lab"><font color="Green" title="E72">A8</font></span></a>, <a class="txt" href="#E10:108_1"><span class="lab"><font color="Green" title="E74">A10</font></span></a>, <a class="txt" href="#E14:108_1"><span class="lab"><font color="Green" title="E76">A12</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_1_2_3"><span class="kw">suppose </span></a><a NAME="E1:108_1_2_3"/>
<font color="Maroon" title="c15">pp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c1">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c9">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c17">m</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E3:108_1_2_3"/><span class="lab"><font color="Green" title="E79">A16</font></span>: 
<font color="Maroon" title="c15">pp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c17">m</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T10" title="NAT_1:th.10">NAT_1:10</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c18">m</font> = <font color="Maroon" title="c17">m</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/><a NAME="E5:108_1_2_3"/>
<font color="Maroon" title="c15">pp</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c18">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:108_1_2_3"><span class="lab"><font color="Green" title="E79">A16</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E6:108_1_2_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:108"><span class="lab"><font color="Green" title="E68">A4</font></span></a>, <a class="txt" href="#E3:108"><span class="lab"><font color="Green" title="E67">A2</font></span></a>, <a class="txt" href="#E2:108_1"><span class="lab"><font color="Green" title="E71">A7</font></span></a>, <a class="txt" href="#E7:108_1"><span class="lab"><font color="Green" title="E72">A8</font></span></a>, <a class="txt" href="#E10:108_1"><span class="lab"><font color="Green" title="E74">A10</font></span></a>, <a class="txt" href="#E14:108_1"><span class="lab"><font color="Green" title="E76">A12</font></span></a>, <a class="txt" href="#E18:108_1"><span class="lab"><font color="Green" title="E78">A14</font></span></a>, <a class="txt" href="comput_1.html#E80"><span class="lab"><font color="Green" title="E64">Lm5</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<a NAME="E7:108"/><span class="lab"><font color="Green" title="E70">A17</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T32" title="FUNCT_7:th.32">FUNCT_7:32</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E71">A18</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="108_2"><span class="kw">now </span></a><div class="add"><a NAME="E1:108_2"/><span class="lab"><font color="Green" title="E72">A19</font></span>: 
<font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><span class="lab"><font color="Green" title="E73">A20</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c1">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:108"><span class="lab"><font color="Green" title="E68">A4</font></span></a>, <a class="txt" href="#E7:108"><span class="lab"><font color="Green" title="E70">A17</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:108"><span class="lab"><font color="Green" title="E68">A4</font></span></a>, <a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">FUNCT_7:33</a></span>
;<br/><span class="kw">consider </span><font color="Maroon" title="c9">F</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<span class="p1">(<span class="default"><a href="comput_1.html#K2" title="COMPUT_1:func.2">HFuncs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span><span class="kw"> such that </span><br/><a NAME="E4:108_2"/><span class="lab"><font color="Green" title="E74">A21</font></span>: 
 <a href="comput_1.html#K6" title="COMPUT_1:func.6">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">F</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">i</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E5:108_2"/><span class="lab"><font color="Green" title="E75">A22</font></span>: 
( <font color="Maroon" title="c1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> &amp;  <a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<font color="Maroon" title="c1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">f1</font> implies <font color="Maroon" title="c9">F</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<font color="Maroon" title="c1">i</font></span>)</span></span>)</span> )
 <span class="kw">and </span><br/><a NAME="E6:108_2"/><span class="lab"><font color="Green" title="E76">A23</font></span>: 
( (  not <font color="Maroon" title="c1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> or  not  <a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<font color="Maroon" title="c1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">f1</font> ) implies <font color="Maroon" title="c9">F</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 <span class="kw">and </span><br/><a NAME="E7:108_2"/>
for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">m</font>,<font color="Maroon" title="c9">F</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">m</font>,<font color="Maroon" title="c9">F</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c4">f2</font>]
 <span class="kw">by</span> <span class="lab"><a class="ref" href="comput_1.html#D13" target="_self" title="COMPUT_1:def.13">Def13</a></span>;<br/><a NAME="E8:108_2"/><span class="lab"><font color="Green" title="E77">A24</font></span>: 
<font color="Maroon" title="c7">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="comput_1.html#K6" title="COMPUT_1:func.6">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:108"><span class="lab"><font color="Green" title="E67">A2</font></span></a></span>;<br/><span class="kw">thus </span><a NAME="E9:108_2"/>
( <font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> iff  <a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">f1</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c1">i</font></span>)</span> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="108_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">thus </span><a NAME="E1:108_2_2"/>
( <font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> implies  <a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">f1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:108"><span class="lab"><font color="Green" title="E67">A2</font></span></a>, <a class="txt" href="#E4:108_2"><span class="lab"><font color="Green" title="E74">A21</font></span></a>, <a class="txt" href="#E6:108_2"><span class="lab"><font color="Green" title="E76">A23</font></span></a>, <a class="txt" href="#E2:108_2"><span class="lab"><font color="Green" title="E73">A20</font></span></a>, <a class="ref" href="comput_1.html#T6" target="_self" title="COMPUT_1:th.6">Th6</a>, <a class="ref" href="relat_1.html#T60" title="RELAT_1:th.60">RELAT_1:60</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">f1</font> implies <font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E2:108_2_2"/>
 <a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">f1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span></span><br/>

<span class="kw">then </span> <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> = 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:108"><span class="lab"><font color="Green" title="E68">A4</font></span></a>, <a class="txt" href="#E5:108_2"><span class="lab"><font color="Green" title="E75">A22</font></span></a>, <a class="ref" href="comput_1.html#T6" target="_self" title="COMPUT_1:th.6">Th6</a>, <a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a>, <a class="ref" href="funct_7.html#T32" title="FUNCT_7:th.32">FUNCT_7:32</a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T36" title="FUNCT_7:th.36">FUNCT_7:36</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E4:108_2_2"/>
<font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:108_2"><span class="lab"><font color="Green" title="E77">A24</font></span></a>, <a class="txt" href="#E4:108_2"><span class="lab"><font color="Green" title="E74">A21</font></span></a>, <a class="txt" href="#E2:108_2"><span class="lab"><font color="Green" title="E73">A20</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">assume </span><a NAME="E10:108_2"/>
<font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c1">i</font></span>)</span></span><br/><a NAME="E11:108_2"/><span class="kw">then </span>
<font color="Maroon" title="c9">F</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<font color="Maroon" title="c1">i</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:108"><span class="lab"><font color="Green" title="E67">A2</font></span></a>, <a class="txt" href="#E4:108_2"><span class="lab"><font color="Green" title="E74">A21</font></span></a>, <a class="txt" href="#E5:108_2"><span class="lab"><font color="Green" title="E75">A22</font></span></a>, <a class="txt" href="#E6:108_2"><span class="lab"><font color="Green" title="E76">A23</font></span></a>, <a class="txt" href="#E2:108_2"><span class="lab"><font color="Green" title="E73">A20</font></span></a>, <a class="ref" href="funct_7.html#T36" title="FUNCT_7:th.36">FUNCT_7:36</a>, <a class="ref" href="relat_1.html#T60" title="RELAT_1:th.60">RELAT_1:60</a></span>;<br/><a NAME="E12:108_2"/><span class="kw">then </span>
<font color="Maroon" title="c9">F</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c1">i</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="comput_1.html#T6" target="_self" title="COMPUT_1:th.6">Th6</a></span>;<br/><span class="kw">hence </span><a NAME="E13:108_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c1">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:108_2"><span class="lab"><font color="Green" title="E77">A24</font></span></a>, <a class="txt" href="#E4:108_2"><span class="lab"><font color="Green" title="E74">A21</font></span></a>, <a class="txt" href="#E2:108_2"><span class="lab"><font color="Green" title="E73">A20</font></span></a>, <a class="txt" href="#E1:108_2"><span class="lab"><font color="Green" title="E72">A19</font></span></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">set </span><font color="Maroon" title="c9">pm1</font> = <font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">pm</font> = <font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c11">pc</font> = <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>;<br/>
<a NAME="E9:108"/><span class="lab"><font color="Green" title="E72">A25</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c3">f1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E10:108"/><span class="kw">then </span><span class="lab"><font color="Green" title="E73">A26</font></span>: 
<font color="Maroon" title="c7">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">rngG</font> = <font color="Maroon" title="c8">rngG</font> as  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="funct_1.html#V4" title="FUNCT_1:attr.4">functional</a>   <a href="comput_1.html#V1" title="COMPUT_1:attr.1">compatible</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/>
<a NAME="E12:108"/><span class="lab"><font color="Green" title="E74">A27</font></span>: 
<font color="Maroon" title="c7">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:108"><span class="lab"><font color="Green" title="E72">A25</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<span class="kw">thus </span><a NAME="E13:108"/>
( <font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> iff  <a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">f1</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> implies <span class="p1">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c1">i</font></span>)</span> ) &amp; ( <font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> implies ( <font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">f2</font> ) ) &amp; ( <font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">f2</font> implies <font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> ) &amp; ( <font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> implies <span class="p1">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="108_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">thus </span><a NAME="E1:108_3"/>
( <font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> implies  <a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">f1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:108"><span class="lab"><font color="Green" title="E69">A5</font></span></a>, <a class="txt" href="#E8:108"><span class="lab"><font color="Green" title="E71">A18</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">f1</font> implies <font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E2:108_3"/><span class="lab"><font color="Green" title="E75">A28</font></span>: 
 <a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">f1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span></span><br/>

<a NAME="E3:108_3"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b1">f</font></span>)</span> where <font color="Olive" title="b1">f</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c12">rngG</font> : verum </span>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:108"><span class="lab"><font color="Green" title="E74">A27</font></span></a></span>;<br/>
<a NAME="E4:108_3"/><span class="kw">then </span>
<font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b1">f</font></span>)</span> where <font color="Olive" title="b1">f</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c12">rngG</font> : verum </span>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:108"><span class="lab"><font color="Green" title="E71">A18</font></span></a>, <a class="txt" href="#E2:108_3"><span class="lab"><font color="Green" title="E75">A28</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:108_3"/>
<font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:108"><span class="lab"><font color="Green" title="E66">A1</font></span></a>, <a class="ref" href="comput_1.html#T15" target="_self" title="COMPUT_1:th.15">Th15</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> implies ( <font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">f2</font> ) ) &amp; ( <font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">f2</font> implies <font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> ) &amp; ( <font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> implies <span class="p1">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> ) )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:108_4"/><span class="lab"><font color="Green" title="E75">A29</font></span>: 
<font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c1">i</font></span>)</span></span><br/><a NAME="E2:108_4"/><span class="kw">then </span>
<font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:108"><span class="lab"><font color="Green" title="E69">A5</font></span></a></span>;<br/><a NAME="E3:108_4"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c12">rngG</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:108"><span class="lab"><font color="Green" title="E74">A27</font></span></a>, <a class="ref" href="comput_1.html#T16" target="_self" title="COMPUT_1:th.16">Th16</a></span>;<br/><span class="kw">hence </span><a NAME="E4:108_4"/>
<span class="p1">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c1">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:108"><span class="lab"><font color="Green" title="E66">A1</font></span></a>, <a class="txt" href="#E6:108"><span class="lab"><font color="Green" title="E69">A5</font></span></a>, <a class="txt" href="#E8:108"><span class="lab"><font color="Green" title="E71">A18</font></span></a>, <a class="txt" href="#E1:108_4"><span class="lab"><font color="Green" title="E75">A29</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div>

<a NAME="E15:108"/><span class="lab"><font color="Green" title="E75">A30</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T32" title="FUNCT_7:th.32">FUNCT_7:32</a></span>;<br/>
<a NAME="E16:108"/><span class="lab"><font color="Green" title="E76">A31</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T36" title="FUNCT_7:th.36">FUNCT_7:36</a></span>;<br/>
<a NAME="E17:108"/><span class="lab"><font color="Green" title="E77">A32</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T36" title="FUNCT_7:th.36">FUNCT_7:36</a></span>;<br/>
<a NAME="E18:108"/><span class="lab"><font color="Green" title="E78">A33</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T32" title="FUNCT_7:th.32">FUNCT_7:32</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E79">A34</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="108_5"><span class="kw">now </span></a><div class="add"><span class="lab"><font color="Green" title="E80">A35</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c1">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:108"><span class="lab"><font color="Green" title="E68">A4</font></span></a>, <a class="txt" href="#E18:108"><span class="lab"><font color="Green" title="E78">A33</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>
<br/>.= 
<font color="Maroon" title="c2">m</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:108"><span class="lab"><font color="Green" title="E68">A4</font></span></a>, <a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">FUNCT_7:33</a></span>
;<br/><span class="kw">consider </span><font color="Maroon" title="c13">F</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<span class="p1">(<span class="default"><a href="comput_1.html#K2" title="COMPUT_1:func.2">HFuncs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span><span class="kw"> such that </span><br/><a NAME="E3:108_5"/><span class="lab"><font color="Green" title="E81">A36</font></span>: 
 <a href="comput_1.html#K6" title="COMPUT_1:func.6">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">F</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">i</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:108_5"/><span class="lab"><font color="Green" title="E82">A37</font></span>: 
( <font color="Maroon" title="c1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span> &amp;  <a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span>,<font color="Maroon" title="c1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">f1</font> implies <font color="Maroon" title="c13">F</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span>,<font color="Maroon" title="c1">i</font></span>)</span></span>)</span> )
 <span class="kw">and </span><br/><a NAME="E5:108_5"/><span class="lab"><font color="Green" title="E83">A38</font></span>: 
( (  not <font color="Maroon" title="c1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span> or  not  <a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span>,<font color="Maroon" title="c1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">f1</font> ) implies <font color="Maroon" title="c13">F</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 <span class="kw">and </span><br/><a NAME="E6:108_5"/><span class="lab"><font color="Green" title="E84">A39</font></span>: 
for <font color="Olive" title="b1">M</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">M</font>,<font color="Maroon" title="c13">F</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">M</font>,<font color="Maroon" title="c13">F</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">M</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c4">f2</font>]
 <span class="kw">by</span> <span class="lab"><a class="ref" href="comput_1.html#D13" target="_self" title="COMPUT_1:def.13">Def13</a></span>;<br/><a NAME="E7:108_5"/><span class="lab"><font color="Green" title="E85">A40</font></span>: 
<font color="Maroon" title="c7">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="comput_1.html#K6" title="COMPUT_1:func.6">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:108"><span class="lab"><font color="Green" title="E67">A2</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c14">F1</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<span class="p1">(<span class="default"><a href="comput_1.html#K2" title="COMPUT_1:func.2">HFuncs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span>)</span><span class="kw"> such that </span><br/><a NAME="E9:108_5"/><span class="lab"><font color="Green" title="E86">A41</font></span>: 
 <a href="comput_1.html#K6" title="COMPUT_1:func.6">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">i</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E10:108_5"/><span class="lab"><font color="Green" title="E87">A42</font></span>: 
( <font color="Maroon" title="c1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> &amp;  <a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">f1</font> implies <font color="Maroon" title="c14">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p2">(<span class="default"><a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">i</font></span>)</span></span>)</span> )
 <span class="kw">and </span><br/><a NAME="E11:108_5"/><span class="lab"><font color="Green" title="E88">A43</font></span>: 
( (  not <font color="Maroon" title="c1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> or  not  <a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">f1</font> ) implies <font color="Maroon" title="c14">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 <span class="kw">and </span><br/><a NAME="E12:108_5"/><span class="lab"><font color="Green" title="E89">A44</font></span>: 
for <font color="Olive" title="b1">M</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">M</font>,<font color="Maroon" title="c14">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">M</font>,<font color="Maroon" title="c14">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">M</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c4">f2</font>]
 <span class="kw">by</span> <span class="lab"><a class="ref" href="comput_1.html#D13" target="_self" title="COMPUT_1:def.13">Def13</a></span>;<br/><a NAME="E13:108_5"/><span class="lab"><font color="Green" title="E90">A45</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c14">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:108"><span class="lab"><font color="Green" title="E68">A4</font></span></a>, <a class="txt" href="#E10:108_5"><span class="lab"><font color="Green" title="E87">A42</font></span></a>, <a class="txt" href="#E11:108_5"><span class="lab"><font color="Green" title="E88">A43</font></span></a>, <a class="txt" href="#E12:108_5"><span class="lab"><font color="Green" title="E89">A44</font></span></a>, <a class="txt" href="comput_1.html#E78"><span class="lab"><font color="Green" title="E62">Lm3</font></span></a></span>;<br/><a NAME="E14:108_5"/><span class="lab"><font color="Green" title="E91">A46</font></span>: 
<font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E15:108_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E92">A47</font></span>: 
<font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p4"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c14">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></span>;<br/><a NAME="E16:108_5"/><span class="lab"><font color="Green" title="E93">A48</font></span>: 
<font color="Maroon" title="c7">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="comput_1.html#K6" title="COMPUT_1:func.6">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:108"><span class="lab"><font color="Green" title="E67">A2</font></span></a></span>;<br/><span class="lab"><font color="Green" title="E94">A49</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c1">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:108"><span class="lab"><font color="Green" title="E68">A4</font></span></a>, <a class="txt" href="#E15:108"><span class="lab"><font color="Green" title="E75">A30</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>
<br/>.= 
<font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:108"><span class="lab"><font color="Green" title="E68">A4</font></span></a>, <a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">FUNCT_7:33</a></span>
;<br/><a NAME="E18:108_5"/><span class="lab"><font color="Green" title="E95">A50</font></span>: 
<font color="Maroon" title="c14">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">F</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c2">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:108_5"><span class="lab"><font color="Green" title="E87">A42</font></span></a>, <a class="txt" href="#E11:108_5"><span class="lab"><font color="Green" title="E88">A43</font></span></a>, <a class="txt" href="#E12:108_5"><span class="lab"><font color="Green" title="E89">A44</font></span></a>, <a class="txt" href="#E4:108_5"><span class="lab"><font color="Green" title="E82">A37</font></span></a>, <a class="txt" href="#E5:108_5"><span class="lab"><font color="Green" title="E83">A38</font></span></a>, <a class="txt" href="#E6:108_5"><span class="lab"><font color="Green" title="E84">A39</font></span></a>, <a class="txt" href="comput_1.html#E77"><span class="lab"><font color="Green" title="E61">Lm2</font></span></a></span>;<br/><a NAME="E19:108_5"/><span class="lab"><font color="Green" title="E96">A51</font></span>: 
not <font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c2">m</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:108"><span class="lab"><font color="Green" title="E68">A4</font></span></a>, <a class="txt" href="#E10:108_5"><span class="lab"><font color="Green" title="E87">A42</font></span></a>, <a class="txt" href="#E11:108_5"><span class="lab"><font color="Green" title="E88">A43</font></span></a>, <a class="txt" href="#E12:108_5"><span class="lab"><font color="Green" title="E89">A44</font></span></a>, <a class="txt" href="comput_1.html#E78"><span class="lab"><font color="Green" title="E62">Lm3</font></span></a></span>;<br/><span class="kw">thus </span><a NAME="E20:108_5"/><span class="lab"><font color="Green" title="E97">A52</font></span>: 
( <font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> iff ( <font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">f2</font> ) )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p0">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p0">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="108_5_3"><span class="kw">proof </span></a><div class="add">

<div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">f2</font> implies <font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:108_5_3_1"/><span class="lab"><font color="Green" title="E98">A53</font></span>: 
<font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">f2</font> )</span><br/><a NAME="E2:108_5_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E99">A54</font></span>: 
<font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:108"><span class="lab"><font color="Green" title="E67">A2</font></span></a>, <a class="txt" href="#E9:108_5"><span class="lab"><font color="Green" title="E86">A41</font></span></a>, <a class="txt" href="#E17:108_5"><span class="lab"><font color="Green" title="E94">A49</font></span></a></span>;<br/><span class="kw">assume </span><a NAME="E3:108_5_3_1"/><span class="lab"><font color="Green" title="E100">A55</font></span>: 
(  not <font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span></span>)</span> or  not <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">f2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_5_3_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:108_5_3_1_1"/>
(  not <font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span></span>)</span> or  not <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">f2</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:108_5_3_1"><span class="lab"><font color="Green" title="E100">A55</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_5_3_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:108_5_3_1_1_1"/>
not <font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:108_5_3_1_1_1"/><span class="kw">then </span>
not <font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c2">m</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:108"><span class="lab"><font color="Green" title="E67">A2</font></span></a>, <a class="txt" href="#E3:108_5"><span class="lab"><font color="Green" title="E81">A36</font></span></a>, <a class="txt" href="#E1:108_5"><span class="lab"><font color="Green" title="E80">A35</font></span></a>, <a class="txt" href="#E18:108_5"><span class="lab"><font color="Green" title="E95">A50</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:108_5_3_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:108"><span class="lab"><font color="Green" title="E77">A32</font></span></a>, <a class="txt" href="#E12:108_5"><span class="lab"><font color="Green" title="E89">A44</font></span></a>, <a class="txt" href="#E19:108_5"><span class="lab"><font color="Green" title="E96">A51</font></span></a>, <a class="txt" href="#E2:108_5_3_1"><span class="lab"><font color="Green" title="E99">A54</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_5_3_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:108_5_3_1_1_2"/>
not <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">f2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:108_5_3_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:108"><span class="lab"><font color="Green" title="E77">A32</font></span></a>, <a class="txt" href="#E7:108_5"><span class="lab"><font color="Green" title="E85">A40</font></span></a>, <a class="txt" href="#E9:108_5"><span class="lab"><font color="Green" title="E86">A41</font></span></a>, <a class="txt" href="#E12:108_5"><span class="lab"><font color="Green" title="E89">A44</font></span></a>, <a class="txt" href="#E17:108_5"><span class="lab"><font color="Green" title="E94">A49</font></span></a>, <a class="txt" href="#E13:108_5"><span class="lab"><font color="Green" title="E90">A45</font></span></a>, <a class="txt" href="#E19:108_5"><span class="lab"><font color="Green" title="E96">A51</font></span></a>, <a class="txt" href="#E1:108_5_3_1"><span class="lab"><font color="Green" title="E98">A53</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div>
<span class="kw">end;</span></div>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:108_5_3"/><span class="lab"><font color="Green" title="E98">A56</font></span>: 
<font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E3:108_5_3"/><span class="lab"><font color="Green" title="E99">A57</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">f2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span><br/>

<a NAME="E4:108_5_3"/>
<font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E5:108_5_3"/><span class="kw">then </span>
<font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p4"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c14">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></span>;<br/>
<a NAME="E6:108_5_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E100">A58</font></span>: 
<font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c2">m</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">f2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p5"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c14">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E7:108_5_3"/>
<font color="Maroon" title="c14">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p4"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c14">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:108"><span class="lab"><font color="Green" title="E68">A4</font></span></a>, <a class="txt" href="#E15:108"><span class="lab"><font color="Green" title="E75">A30</font></span></a>, <a class="txt" href="#E17:108"><span class="lab"><font color="Green" title="E77">A32</font></span></a>, <a class="txt" href="#E16:108"><span class="lab"><font color="Green" title="E76">A31</font></span></a>, <a class="txt" href="#E7:108_5"><span class="lab"><font color="Green" title="E85">A40</font></span></a>, <a class="txt" href="#E16:108_5"><span class="lab"><font color="Green" title="E93">A48</font></span></a>, <a class="txt" href="#E9:108_5"><span class="lab"><font color="Green" title="E86">A41</font></span></a>, <a class="txt" href="#E12:108_5"><span class="lab"><font color="Green" title="E89">A44</font></span></a>, <a class="txt" href="#E3:108_5"><span class="lab"><font color="Green" title="E81">A36</font></span></a>, <a class="txt" href="#E17:108_5"><span class="lab"><font color="Green" title="E94">A49</font></span></a>, <a class="txt" href="#E1:108_5"><span class="lab"><font color="Green" title="E80">A35</font></span></a>, <a class="txt" href="#E18:108_5"><span class="lab"><font color="Green" title="E95">A50</font></span></a>, <a class="txt" href="#E13:108_5"><span class="lab"><font color="Green" title="E90">A45</font></span></a>, <a class="txt" href="#E2:108_5_3"><span class="lab"><font color="Green" title="E98">A56</font></span></a>, <a class="txt" href="#E3:108_5_3"><span class="lab"><font color="Green" title="E99">A57</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:108_5_3"/>
<font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:108_5"><span class="lab"><font color="Green" title="E85">A40</font></span></a>, <a class="txt" href="#E9:108_5"><span class="lab"><font color="Green" title="E86">A41</font></span></a>, <a class="txt" href="#E17:108_5"><span class="lab"><font color="Green" title="E94">A49</font></span></a>, <a class="txt" href="#E6:108_5_3"><span class="lab"><font color="Green" title="E100">A58</font></span></a>, <a class="ref" href="funct_4.html#D1" title="FUNCT_4:def.1">FUNCT_4:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">assume </span><a NAME="E21:108_5"/><span class="lab"><font color="Green" title="E98">A59</font></span>: 
<font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p0">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p0">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span><br/><a NAME="E22:108_5"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c14">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">f2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:108"><span class="lab"><font color="Green" title="E77">A32</font></span></a>, <a class="txt" href="#E7:108_5"><span class="lab"><font color="Green" title="E85">A40</font></span></a>, <a class="txt" href="#E9:108_5"><span class="lab"><font color="Green" title="E86">A41</font></span></a>, <a class="txt" href="#E12:108_5"><span class="lab"><font color="Green" title="E89">A44</font></span></a>, <a class="txt" href="#E17:108_5"><span class="lab"><font color="Green" title="E94">A49</font></span></a>, <a class="txt" href="#E20:108_5"><span class="lab"><font color="Green" title="E97">A52</font></span></a></span>;<br/><a NAME="E23:108_5"/><span class="kw">then </span>
<font color="Maroon" title="c14">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p4"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c14">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:108"><span class="lab"><font color="Green" title="E68">A4</font></span></a>, <a class="txt" href="#E15:108"><span class="lab"><font color="Green" title="E75">A30</font></span></a>, <a class="txt" href="#E17:108"><span class="lab"><font color="Green" title="E77">A32</font></span></a>, <a class="txt" href="#E16:108"><span class="lab"><font color="Green" title="E76">A31</font></span></a>, <a class="txt" href="#E16:108_5"><span class="lab"><font color="Green" title="E93">A48</font></span></a>, <a class="txt" href="#E12:108_5"><span class="lab"><font color="Green" title="E89">A44</font></span></a>, <a class="txt" href="#E3:108_5"><span class="lab"><font color="Green" title="E81">A36</font></span></a>, <a class="txt" href="#E1:108_5"><span class="lab"><font color="Green" title="E80">A35</font></span></a>, <a class="txt" href="#E18:108_5"><span class="lab"><font color="Green" title="E95">A50</font></span></a>, <a class="txt" href="#E20:108_5"><span class="lab"><font color="Green" title="E97">A52</font></span></a>, <a class="txt" href="#E21:108_5"><span class="lab"><font color="Green" title="E98">A59</font></span></a></span>;<br/><span class="kw">hence </span><span class="p1">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p4"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c14">F1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c2">m</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:108_5"><span class="lab"><font color="Green" title="E85">A40</font></span></a>, <a class="txt" href="#E9:108_5"><span class="lab"><font color="Green" title="E86">A41</font></span></a>, <a class="txt" href="#E17:108_5"><span class="lab"><font color="Green" title="E94">A49</font></span></a>, <a class="txt" href="#E15:108_5"><span class="lab"><font color="Green" title="E92">A47</font></span></a>, <a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></span>
<br/>.= 
<font color="Maroon" title="c4">f2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p0">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p0">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:108"><span class="lab"><font color="Green" title="E77">A32</font></span></a>, <a class="txt" href="#E7:108_5"><span class="lab"><font color="Green" title="E85">A40</font></span></a>, <a class="txt" href="#E9:108_5"><span class="lab"><font color="Green" title="E86">A41</font></span></a>, <a class="txt" href="#E17:108_5"><span class="lab"><font color="Green" title="E94">A49</font></span></a>, <a class="txt" href="#E13:108_5"><span class="lab"><font color="Green" title="E90">A45</font></span></a>, <a class="txt" href="#E14:108_5"><span class="lab"><font color="Green" title="E91">A46</font></span></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">thus </span><a NAME="E20:108"/>
( <font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> iff ( <font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">f2</font> ) )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> implies <span class="p1">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="108_6"><span class="kw">proof </span></a><div class="add">

<div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">f2</font> implies <font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:108_6_1"/><span class="lab"><font color="Green" title="E80">A60</font></span>: 
<font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">f2</font> )</span><br/><a NAME="E2:108_6_1"/>
<font color="Maroon" title="c7">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:108"><span class="lab"><font color="Green" title="E72">A25</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/><a NAME="E3:108_6_1"/><span class="kw">then </span>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b1">f</font></span>)</span> where <font color="Olive" title="b1">f</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c12">rngG</font> : verum </span>}</span> 
 ;<br/><a NAME="E4:108_6_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b1">f</font></span>)</span> where <font color="Olive" title="b1">f</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c12">rngG</font> : verum </span>}</span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:108"><span class="lab"><font color="Green" title="E69">A5</font></span></a>, <a class="txt" href="#E19:108"><span class="lab"><font color="Green" title="E79">A34</font></span></a>, <a class="txt" href="#E1:108_6_1"><span class="lab"><font color="Green" title="E80">A60</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/><span class="kw">hence </span><a NAME="E5:108_6_1"/>
<font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:108"><span class="lab"><font color="Green" title="E66">A1</font></span></a>, <a class="ref" href="comput_1.html#T15" target="_self" title="COMPUT_1:th.15">Th15</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">f2</font></span><br/><a NAME="E6:108_6_1"/><span class="lab"><font color="Green" title="E81">A61</font></span>: 
<font color="Maroon" title="c7">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:108"><span class="lab"><font color="Green" title="E72">A25</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/><a NAME="E7:108_6_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:108"><span class="lab"><font color="Green" title="E68">A4</font></span></a>, <a class="txt" href="#E3:108"><span class="lab"><font color="Green" title="E67">A2</font></span></a>, <a class="txt" href="comput_1.html#E79"><span class="lab"><font color="Green" title="E63">Lm4</font></span></a></span>;<br/><a NAME="E8:108_6_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c12">rngG</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:108"><span class="lab"><font color="Green" title="E69">A5</font></span></a>, <a class="txt" href="#E19:108"><span class="lab"><font color="Green" title="E79">A34</font></span></a>, <a class="txt" href="#E1:108_6_1"><span class="lab"><font color="Green" title="E80">A60</font></span></a>, <a class="txt" href="#E6:108_6_1"><span class="lab"><font color="Green" title="E81">A61</font></span></a>, <a class="ref" href="comput_1.html#T16" target="_self" title="COMPUT_1:th.16">Th16</a></span>;<br/><span class="kw">hence </span><a NAME="E9:108_6_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">f2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:108"><span class="lab"><font color="Green" title="E66">A1</font></span></a>, <a class="txt" href="#E6:108"><span class="lab"><font color="Green" title="E69">A5</font></span></a>, <a class="txt" href="#E19:108"><span class="lab"><font color="Green" title="E79">A34</font></span></a>, <a class="txt" href="#E1:108_6_1"><span class="lab"><font color="Green" title="E80">A60</font></span></a>, <a class="ref" href="funct_7.html#T36" title="FUNCT_7:th.36">FUNCT_7:36</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:108_6"/><span class="lab"><font color="Green" title="E80">A62</font></span>: 
<font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E3:108_6"/><span class="lab"><font color="Green" title="E81">A63</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">f2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span></span><br/>

<a NAME="E4:108_6"/><span class="lab"><font color="Green" title="E82">A64</font></span>: 
<font color="Maroon" title="c7">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:108"><span class="lab"><font color="Green" title="E72">A25</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<a NAME="E5:108_6"/>
<font color="Maroon" title="c7">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:108"><span class="lab"><font color="Green" title="E72">A25</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<a NAME="E6:108_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E83">A65</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b1">f</font></span>)</span> where <font color="Olive" title="b1">f</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c12">rngG</font> : verum </span>}</span> 
 
;<br/>
<a NAME="E7:108_6"/><span class="lab"><font color="Green" title="E84">A66</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:108"><span class="lab"><font color="Green" title="E68">A4</font></span></a>, <a class="txt" href="#E3:108"><span class="lab"><font color="Green" title="E67">A2</font></span></a>, <a class="txt" href="comput_1.html#E79"><span class="lab"><font color="Green" title="E63">Lm4</font></span></a></span>;<br/>
<a NAME="E8:108_6"/>
<font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:108"><span class="lab"><font color="Green" title="E69">A5</font></span></a>, <a class="txt" href="#E2:108_6"><span class="lab"><font color="Green" title="E80">A62</font></span></a></span>;<br/>
<a NAME="E9:108_6"/><span class="kw">then </span>
<font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b1">f</font></span>)</span> where <font color="Olive" title="b1">f</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c12">rngG</font> : verum </span>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:108"><span class="lab"><font color="Green" title="E66">A1</font></span></a>, <a class="txt" href="#E17:108"><span class="lab"><font color="Green" title="E77">A32</font></span></a>, <a class="txt" href="#E19:108"><span class="lab"><font color="Green" title="E79">A34</font></span></a>, <a class="txt" href="#E3:108_6"><span class="lab"><font color="Green" title="E81">A63</font></span></a>, <a class="txt" href="#E6:108_6"><span class="lab"><font color="Green" title="E83">A65</font></span></a>, <a class="txt" href="#E4:108_6"><span class="lab"><font color="Green" title="E82">A64</font></span></a>, <a class="txt" href="#E7:108_6"><span class="lab"><font color="Green" title="E84">A66</font></span></a>, <a class="ref" href="comput_1.html#T16" target="_self" title="COMPUT_1:th.16">Th16</a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:108_6"/>
<font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:108"><span class="lab"><font color="Green" title="E66">A1</font></span></a>, <a class="ref" href="comput_1.html#T15" target="_self" title="COMPUT_1:th.15">Th15</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E21:108"/><span class="lab"><font color="Green" title="E80">A67</font></span>: 
<font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span><br/>

<a NAME="E22:108"/><span class="lab"><font color="Green" title="E81">A68</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:108"><span class="lab"><font color="Green" title="E68">A4</font></span></a>, <a class="txt" href="#E3:108"><span class="lab"><font color="Green" title="E67">A2</font></span></a>, <a class="txt" href="comput_1.html#E79"><span class="lab"><font color="Green" title="E63">Lm4</font></span></a></span>;<br/>
<a NAME="E23:108"/><span class="kw">then </span><span class="lab"><font color="Green" title="E82">A69</font></span>: 
<span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c12">rngG</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:108"><span class="lab"><font color="Green" title="E69">A5</font></span></a>, <a class="txt" href="#E21:108"><span class="lab"><font color="Green" title="E80">A67</font></span></a>, <a class="txt" href="#E10:108"><span class="lab"><font color="Green" title="E73">A26</font></span></a>, <a class="ref" href="comput_1.html#T16" target="_self" title="COMPUT_1:th.16">Th16</a></span>;<br/>
<a NAME="E24:108"/>
<span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c12">rngG</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:108"><span class="lab"><font color="Green" title="E69">A5</font></span></a>, <a class="txt" href="#E19:108"><span class="lab"><font color="Green" title="E79">A34</font></span></a>, <a class="txt" href="#E21:108"><span class="lab"><font color="Green" title="E80">A67</font></span></a>, <a class="txt" href="#E10:108"><span class="lab"><font color="Green" title="E73">A26</font></span></a>, <a class="txt" href="#E22:108"><span class="lab"><font color="Green" title="E81">A68</font></span></a>, <a class="ref" href="comput_1.html#T16" target="_self" title="COMPUT_1:th.16">Th16</a></span>;<br/>
<span class="kw">hence </span><a NAME="E25:108"/>
<span class="p1">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="comput_1.html#K7" title="COMPUT_1:func.7">primrec</a> <font color="Maroon" title="c3">f1</font>,<font color="Maroon" title="c4">f2</font>,<font color="Maroon" title="c1">i</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">p</font> <a href="comput_1.html#K8" title="COMPUT_1:func.8">+*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">m</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:108"><span class="lab"><font color="Green" title="E66">A1</font></span></a>, <a class="txt" href="#E6:108"><span class="lab"><font color="Green" title="E69">A5</font></span></a>, <a class="txt" href="#E19:108"><span class="lab"><font color="Green" title="E79">A34</font></span></a>, <a class="txt" href="#E21:108"><span class="lab"><font color="Green" title="E80">A67</font></span></a>, <a class="txt" href="#E23:108"><span class="lab"><font color="Green" title="E82">A69</font></span></a>, <a class="ref" href="funct_7.html#T36" title="FUNCT_7:th.36">FUNCT_7:36</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
